Date: Jan 13, 2013 5:20 AM
Author: Zaljohar@gmail.com
Subject: A intuitive notion of set size.

For any two sets A,B:
[1] A is bigger than B iff |A| > |B| Or A,B are sets of naturals and
there exist a non empty set C of naturals such that for every element
n of C:
|A(n)| > |B(n)| and |A(n*)| / |B(n*)| >= |A(n)| / |B(n)|.
where X(n) = {y| y in X & y <' n};
<' stands for natural strict smaller than relation;
| | stands for cardinality defined after Cantor's.
n* stands for the immediate successor of n in C with respect to
natural succession.
[2] A is smaller than B iff B is bigger than A.
[3] A is equinumerous to B iff ~ A bigger than B & ~ A smaller than
B.
/

Example:
Let A be the set N of all naturals.
Let B be the set E of all even naturals.
Let C be the set E\{0}, i.e. the set of all even naturals except 0.

Now at each member n of C we do have |N(n)| > |E(n)|
Also we have |N(n*)| / |E(n*)| = |N(n)| / |E(n)|

So N is bigger than E.

-----------------------------------
Definitions given here of 'bigger than' , 'smaller than' and
'equinumerous' in some sense parallel that of set Density. They of
course depart from Cantor's definitions as regards sets of naturals,
and in being so they actually come closer to ordinary intuitions we
have about set sizes that we are familiar with from the finite world,
so for example the set of all naturals is bigger than that of all
evens, the set of all evens is bigger than that of all squares etc..
Actually one can describe interesting sizes of sets of naturals. If we
consider the size of N to be oo, then the set of evens would have the
size oo/2, also we can define any oo/n, oo-n, and n_th root of oo, in
a nice manner comparable to finite set sizes. Obviously Cardinality
cannot achieve that! I do think that somehow this approach can be
extended to cover all sets of reals! And perhaps higher level sets
along the cumulative hierarchy of ZFC as well.

Zuhair